16.3
Résolution de problèmes par équation de Bernoulli
Un débitmètre Venturi est essentiel pour mesurer les débits de fluides dans les canalisations. Il utilise la relation entre la vitesse du fluide et la…
Un compteur Venturi est installé dans un système d’égouts pour mesurer le débit des eaux usées. Si le débit est connu et que les effets visqueux sont négligeables, quelle est la différence de pression indiquée par le manomètre fixé au débitmètre ?
Tout d’abord, calculez les sections transversales du tuyau et de la gorge à l’aide de leurs diamètres respectifs.
Ensuite, à l’aide de l’équation de continuité, déterminez les vitesses aux deux sections en fonction du débit donné.
Étant donné que l’aire et la vitesse sont inversement liées, les eaux usées s’accélèrent lorsqu’elles se déplacent du tuyau de plus grand diamètre vers la plus petite gorge.
Ensuite, appliquez l'équation de Bernoulli, qui relie la différence de pression entre deux points de l'écoulement à la variation de vitesse.
Cette équation montre que lorsque la vitesse augmente lorsque l’eau se déplace du plus grand tuyau vers la plus petite gorge, il y aura une chute de pression correspondante.
Enfin, calculez la différence de pression entre le tuyau et la gorge en substituant les vitesses dans l'équation de Bernoulli.
Le manomètre fixé au débitmètre indiquera la différence de pression finale.
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Q1: How does a Venturi meter measure wastewater flow rate in a sewage system?
A Venturi meter measures flow rate by detecting the pressure difference created when fluid accelerates through a narrower throat section. The meter calculates cross-sectional areas of the pipe and throat, applies the continuity equation to determine velocities at both sections, then uses Bernoulli's equation to relate the pressure drop to velocity changes. The pressure gauge attached to the flowmeter indicates this pressure difference, which reflects the flow rate.
Q2: What is the relationship between velocity and pressure in a Venturi meter?
As wastewater flows from the larger-diameter pipe into the smaller throat, it accelerates due to the continuity equation, which requires constant volumetric flow rate. This velocity increase causes a corresponding pressure drop, as described by Bernoulli's equation for flow along a streamline. The pressure difference between the pipe and throat is inversely related to velocity: higher velocity at the throat means lower pressure at that location.
Q3: Why must you calculate cross-sectional areas before applying Bernoulli's equation to a Venturi meter?
Cross-sectional areas are essential because they determine the velocities at each section through the continuity equation. Since volumetric flow rate remains constant for incompressible fluid, the velocity is inversely proportional to area. Knowing both areas allows you to calculate the exact velocities needed to substitute into Bernoulli's equation and determine the pressure difference.
Q4: How does the continuity equation apply to flow through a Venturi meter?
The continuity equation states that volumetric flow rate remains constant along the pipeline for incompressible fluid. This means the flow rate at the pipe equals the flow rate at the throat. Since area and velocity are inversely related, the smaller throat area forces the wastewater to accelerate, creating the velocity difference that Bernoulli's equation uses to calculate pressure drop.
Q5: What does the pressure gauge reading on a Venturi meter actually indicate?
The pressure gauge indicates the pressure difference between the main pipe and the throat section. This pressure drop reflects the increase in velocity as wastewater flows through the narrower throat, consistent with energy conservation principles. The magnitude of this pressure difference directly correlates to the flow rate, allowing operators to determine wastewater flow from a single gauge reading.
Q6: Why is viscous effects being negligible important for Venturi meter calculations?
Neglecting viscous effects allows you to apply Bernoulli's equation directly, which assumes ideal fluid behavior without energy loss to friction. This simplification is valid when viscous forces are small compared to inertial forces. Under this assumption, all pressure changes result solely from velocity changes, making the relationship between pressure difference and flow rate straightforward and reliable.
Q7: How do you calculate the pressure difference using Bernoulli's equation in a Venturi meter?
Bernoulli's equation relates pressure and velocity in an ideal fluid: P1 + 0.5ρV1² = P2 + 0.5ρV2². Rearranging gives the pressure difference: P1 - P2 = 0.5ρ(V2² - V1²). After calculating velocities V1 and V2 from the continuity equation and known flow rate, substitute them along with fluid density into this formula to find the pressure difference the gauge will display.